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SOME QUESTIONS OF UNIFORMITY IN ALGORITHMIC RANDOMNESS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  12 August 2021

LAURENT BIENVENU ,
BARBARA F. CSIMA  and
MATTHEW HARRISON-TRAINOR
LAURENT BIENVENU
Affiliation:
LABORATOIRE BORDELAIS DE RECHERCHE EN INFORMATIQUE CNRS, UNIVERSITÉ DE BORDEAUX BORDEAUX INP TALENCE, FRANCE E-mail: laurent.bienvenu@u-bordeaux.fr
BARBARA F. CSIMA
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO WATERLOO, ON, CANADA E-mail: csima@uwaterloo.ca
MATTHEW HARRISON-TRAINOR
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGAN ANN ARBOR, MI, USA E-mail: matthhar@umich.edu
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Abstract

The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $, a universal prefix-free machine U whose halting probability is $\alpha $. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $, one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.

Keywords

algorithmic randomnessKolmogorov complexity

MSC classification

Primary: 03D32: Algorithmic randomness and dimension

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1612 - 1631
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Copyright
© Association for Symbolic Logic 2021

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